Big Bang nucleosynthesis

Builds on: Sections 13.4, 13.5 and 13.7
Reading time: 35 min
Exercises: 3
Olympiad problems: 1

About a quarter of the mass of ordinary matter is helium, and it is nearly the same everywhere we look, in old stars, in young ones and in small galaxies whose gas has barely been touched by stars. The stars cannot have made it all. Burning that much hydrogen into helium over the age of the Universe would have made galaxies far brighter than they are. Most of the helium was made in the first few minutes after the Big Bang, when the whole Universe was as hot as the centre of a star. In this section we follow that process step by step, as IOAA 2025 did, and find why the answer comes out at a quarter.

13.8.1Neutrons and protons

Before you readMake a guess first

A neutron is slightly heavier than a proton. When the Universe was very hot, which were more common?

In the first second, reactions with electrons, positrons and neutrinos kept turning neutrons into protons and back again. The neutron is heavier by $(m_{n} - m_{p}) c^{2} = 1.293$ MeV, so in this equilibrium the neutrons were outnumbered by the Boltzmann factor $e^{- Δ E / k T}$ of that energy difference $Δ E$ , the same factor that sets how many atoms sit in an excited level of a hot gas. As the Universe cooled, the reactions slowed until, at about $k T = 0.8$ MeV and $t = 1.7$ s, they could no longer keep up with the expansion. The ratio then froze. This moment is called freeze-out.

ResultThe balance of neutrons and protons(13.15)

\frac{N_{n}}{N_{p}} = {(\frac{m_{n}}{m_{p}})}^{3 / 2} \exp (- \frac{(m_{n} - m_{p}) c^{2}}{k T})

$N_{n}, N_{p}$	the numbers of free neutrons and protons in each cubic metre
$m_{n}, m_{p}$	their masses
$T$	the temperature, while the reactions keep the two in balance

At freeze-out,

k T = 800

keV, the ratio is 0.199, so neutrons make up

X_{n} = N_{n} / (N_{n} + N_{p}) = 0.166

, about one in six, of all the neutrons and protons. The factor

(m_{n} / m_{p})^{3 / 2}

is only 1.002.

13.8.2Waiting for deuterium

After freeze-out, free neutrons decay into protons with a mean life of 880.6 s, a half-life of 610.4 s, so their share slowly falls. They could be saved only by joining protons, first as deuterium, a proton and a neutron bound by 2.22 MeV, and then as helium. Deuterium forms easily, but at first it is broken up again at once. As with recombination (Section 13.7), there are about 1.6 billion photons for each proton or neutron, $1 / η$ , so even when the typical photon is far weaker than 2.22 MeV, the rare energetic ones in the tail of the spectrum split every new deuterium nucleus. This hold-up is called the deuterium bottleneck.

The balance between deuterium and free neutrons follows the Saha equation, which IOAA 2025 gave in a simplified form. It depends on the number of baryons, protons and neutrons, for each photon, $η$ .

ResultDeuterium and free neutrons(13.16)

\frac{N_{D}}{N_{n}} = 6.5 η {(\frac{k T}{m_{n} c^{2}})}^{3 / 2} \exp (\frac{B_{D}}{k T})

$N_{D}$	the number of deuterium nuclei in each cubic metre
$η$	the number of baryons for each photon, about $6.1 \times 10^{- 10}$
$B_{D}$	the binding energy of deuterium, $(m_{n} + m_{p} - m_{D}) c^{2} = 2.22$ MeV

The tiny factor

η

keeps the ratio small until the exponential grows very large, which happens only at

k T \approx 66

keV, about

7.7 \times 10^{8}

K. In the radiation era

T \propto t^{- 1 / 2}

((13.9)), so this is at

t = 1.70 s \times (800 / 66)^{2} \approx 250

s, about four minutes after the Big Bang.

Figure 13.7When deuterium survives

Baryons for each photon

η

Deuterium survives at
Neutron fraction left
Helium fraction

The figure shows the ratio of deuterium to free neutrons from (13.16), on a logarithmic scale, against temperature, which falls to the right as time passes. The ratio rises very steeply as the Universe cools, and N marks where it reaches 1, after which almost all the neutrons are bound and soon go into helium. The slider changes the number of baryons for each photon. More baryons let deuterium survive sooner and hotter, so fewer neutrons have decayed and more helium is made. The figure can be set to , or .

Quick checkHow many neutrons survive the wait

Free neutrons have a mean life of 880.6 s. What fraction of those present at freeze-out, at 1.7 s, are still free at 250 s?

13.8.3Helium

Once deuterium survives, nuclear reactions carry almost all of it, and so almost every neutron, into helium-4, the most tightly bound light nucleus. Each helium nucleus holds two neutrons and two protons, so the mass in helium is twice the mass of the neutrons.

ResultThe helium fraction(13.17)

Y_{p} \approx 2 X_{n}

$Y_{p}$	the fraction of the mass of ordinary matter in helium after the first minutes
$X_{n}$	the share of neutrons among all neutrons and protons when helium forms

This takes the masses of neutrons and protons as equal and ignores the small mass lost in binding. With

X_{n} = 0.166 \times 0.754 = 0.125

Y_{p} = 0.250

, close to the 0.24 to 0.25 measured in the least processed galaxies.

Small amounts of deuterium, helium-3 and lithium-7 are left over. Their amounts depend strongly on $η$ , and the measured amounts of deuterium fix $η$ to about $6 \times 10^{- 10}$ . Since the number of photons is known from the background radiation, this gives the density of ordinary matter, $Ω_{b} \approx 0.05$ . Matter as a whole has $Ω_{m} \approx 0.3$ , so most matter cannot be made of protons and neutrons, which is one of the strongest arguments for dark matter.

Common errorsSection 13.8

The neutron fraction at freeze-out is not the fraction that ends up in helium. About a quarter of the neutrons decay while the Universe waits for deuterium to survive.
The helium fraction is twice the neutron fraction, not equal to it, since each neutron in helium brings a proton with it.
The mean life of the neutron, 880.6 s, goes in the exponential decay $e^{- t / τ}$ . Using the half-life of 610.4 s there needs a factor of $\ln 2$ .
Deuterium does not survive at the temperature that matches its binding energy, about $2.6 \times 10^{10}$ K. The billion or so photons for each baryon hold it back until about $7.7 \times 10^{8}$ K.

ExercisesSection 13.83 questions

13.8.1
If neutrons did not decay at all, what would the helium fraction be, starting from $X_{n} = 0.166$ at freeze-out?
13.8.2
With 25% of the mass in helium and 75% in hydrogen, how many helium atoms are there for each hydrogen atom? Take a helium atom as four times as heavy as a hydrogen atom.
13.8.3
If the Universe had fewer baryons for each photon, how would the helium fraction change?

Olympiad problemsPast IOAA problems, restated1 problem

2025Big Bang nucleosynthesis27 points

Adapted from IOAA 2025 (Mumbai), theory problem 10. We have restated it in our own words and kept the original data. The parts that need the problem's own graphs, 2b and 6, are left out, so the points add up to 27 of the original 35. The solution was last checked on 21 October 2026.

In the radiation era the scale factor grows as $a \propto t^{1 / 2}$ . Until $t_{wk} = 1.70$ s, when $k T_{wk} = 800$ keV, reactions keep neutrons and protons in balance, with the number of each in proportion to $m^{3 / 2} \exp (- m c^{2} / k T)$ . After that, free neutrons decay into protons with a half-life of 610.4 s. Use $m_{n} c^{2} = 939.565$ MeV and $m_{p} c^{2} = 938.272$ MeV.

(1) Find the share of neutrons, $X_{n,wk} = N_{n} / (N_{n} + N_{p})$ , at $t_{wk}$ . [4 points]

(2) The photons keep a blackbody spectrum. Find the power $β$ in $T \propto a^{β}$ . [2 points]

(3) After $t_{wk}$ the formation of deuterium follows the simplified Saha equation of (13.16), with $η = 6.1 \times 10^{- 10}$ and $B_{D} = 2224.6$ keV. Find the temperature $k T_{nuc}$ at which $N_{D} = N_{n}$ , and the time $t_{nuc}$ at which it is reached, assuming that then all free neutrons go into helium at once. [10 points]

(4) Find $X_{n,nuc}$ just before $t_{nuc}$ . [5 points]

(5) Estimate the helium fraction by mass, $Y_{prim}$ , taking $m_{p} = m_{n}$ and $m_{He} = 4 m_{n}$ . [3 points]

(6) If $η$ were smaller, would $N_{D} / N_{n}$ at a given temperature, $T_{nuc}$ , $t_{nuc}$ , $X_{n,nuc}$ and $Y_{prim}$ be larger or smaller? [3 points]

I solved it without the solution

Answer(1) 0.166. (2) $β = - 1$ . (3) $k T_{nuc} = 66.0$ keV at $t_{nuc} = 250$ s. (4) 0.125. (5) 0.250. (6) The ratio is smaller, $T_{nuc}$ lower, $t_{nuc}$ later, and $X_{n,nuc}$ and $Y_{prim}$ smaller.